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Theorem pwex 3959
Description: Power set axiom expressed in class notation. Axiom 4 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Hypothesis
Ref Expression
zfpowcl.1 𝐴 ∈ V
Assertion
Ref Expression
pwex 𝒫 𝐴 ∈ V

Proof of Theorem pwex
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 zfpowcl.1 . 2 𝐴 ∈ V
2 pweq 3389 . . 3 (𝑧 = 𝐴 → 𝒫 𝑧 = 𝒫 𝐴)
32eleq1d 2122 . 2 (𝑧 = 𝐴 → (𝒫 𝑧 ∈ V ↔ 𝒫 𝐴 ∈ V))
4 df-pw 3388 . . 3 𝒫 𝑧 = {𝑦𝑦𝑧}
5 axpow2 3956 . . . . . 6 𝑥𝑦(𝑦𝑧𝑦𝑥)
65bm1.3ii 3905 . . . . 5 𝑥𝑦(𝑦𝑥𝑦𝑧)
7 abeq2 2162 . . . . . 6 (𝑥 = {𝑦𝑦𝑧} ↔ ∀𝑦(𝑦𝑥𝑦𝑧))
87exbii 1512 . . . . 5 (∃𝑥 𝑥 = {𝑦𝑦𝑧} ↔ ∃𝑥𝑦(𝑦𝑥𝑦𝑧))
96, 8mpbir 138 . . . 4 𝑥 𝑥 = {𝑦𝑦𝑧}
109issetri 2581 . . 3 {𝑦𝑦𝑧} ∈ V
114, 10eqeltri 2126 . 2 𝒫 𝑧 ∈ V
121, 3, 11vtocl 2625 1 𝒫 𝐴 ∈ V
Colors of variables: wff set class
Syntax hints:  wb 102  wal 1257   = wceq 1259  wex 1397  wcel 1409  {cab 2042  Vcvv 2574  wss 2944  𝒫 cpw 3386
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-v 2576  df-in 2951  df-ss 2958  df-pw 3388
This theorem is referenced by:  pwexg  3960  p0ex  3966  pp0ex  3967  ord3ex  3968  abexssex  5779  npex  6628  axcnex  6992  pnfxr  8792  mnfxr  8794  ixxex  8868
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